Harmonic measure of arcs of fixed length

Jordan domains $\Omega$ with piece-wise smooth boundaries are treated such that all arcs $\alpha\subset \partial \Omega$ having fixed length $l$, $0<l<\text{length}(\partial \Omega)$, have equal harmonic measures $\omega(z_0,\alpha,\Omega)$ evaluated at some point $z_0\in \Omega$. It is proved that $\Omega$ is a disk centered at $z_0$ if the ratio $l/\text{length}(\partial \Omega)$ is irrational and that $\Omega$ possesses rotational symmetry by some angle $2\pi/n$, $n\ge 2$, around the point $z_0$, if this ratio is rational.